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August 3rd, 2006

one poppy

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Unless lightning strikes, the lower layers of the Earth’s atmosphere contain very small fraction of charged particles. The air is electrically neutral, and indeed is a fairly good insulator. This state of affairs is something to be thankful for.

Imagine what would happen if the air started to carry a tiny ionization fraction. That is, imagine if one out of every million air molecules were stripped of an electron. The ionized air molecules and the electrons would experience an immediate desire to spiral around the Earth’s magnetic field lines. In doing so, they would bash into the surrounding sea of neutral particles and drag them along with their motion.

Bulk motion of charged particles drags magnetic field lines along and vice-versa. Magnetic field lines, however, don’t like being compressed or twisted, and have a tendancy – verging on insistence – to spring back into shape. If the Earth’s atmosphere had a small magnetic field, the jet stream would rapidly wind up the Earth’s magnetic field, which would angrily resist the winding and pull backward on the jetstream. Our normal weather patterns would be thrown into complete and utter disarray.

In the inner regions of a protostellar disk, the temperature is high enough for trace elements such as sodium to lose their outer electrons. This raises the ionization fraction of the disk gas to the point where the ambient magnetic field begins to play an important role. This, in turn, leads to an interesting situation.

Imagine two parcels of disk gas on a circular orbit. Imagine also, that the two parcels are connected by a weak magnetic field line. Next, perturb the leading parcel by pulling backward on it slightly. Such a pull drains orbital energy from the parcel and causes it to drop down to a lower orbit. A lower orbit, however, has a faster rotational velocity. The faster rotational velocity causes the parcel to run forward. This pulls on the magnetic field line, which pulls back, forcing the particle even further down into the gravitational well. Clearly, we have the condition for a runaway situation.

This process, known as the magnetorotational instability was discussed by Chandrasekhar in the late 1950’s, and appears in his monograph on Hydrodynamic and Hydromagnetic Stability, and was brilliantly revived in the context of disks in the early 1990’s by Steve Balbus and John Hawley. The nonlinear outcome of the magnetorotational instability is turbulence in the disk. This turbulence may play an important role in allowing mass to slip down and accrete onto the star.

The magnetorotational instability is a simple consequence of the remarkable fact that self-gravitating systems have a negative heat capacity. Balbus and Hawley completely cleaned up by recognizing the importance of the instability within the context of accretion disk physics. Their 1991 paper has now garnered 960 citations. I’m of the opinion that there may be some similarly useful gems ready to be mined out of several of Chandrasekhar’s more opaque books. In fact, I’m going to put on my mining helmet and stake some claims inside of Ellipsoidal Figures of Equilibrium.

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  1. August 13th, 2006 at 17:38 | #1

    I spent a few years way back in the early 90’s mining EFE with Dong Lai and Stu Shapiro. For example take a look at “Ellipsoidal Figures of Equilibrium: Compressible Models” (http://dx.doi.org/10.1086/191822). So far only 130 citations…

    Chandra’s “The Mathematical Theory of Black Holes” is where I would look next, but that is even more opaque… and probably not so relevant to planetary systems… (Wait a minute! “Planets around Black Holes” … that will be part of something fun I’m organizing for next Summer… http://www.astro.northwestern.edu/Santorini2007/)

    Really nice blog btw!

  2. greg
    August 15th, 2006 at 04:43 | #2

    Fred,

    Thanks!

    130 citations is not at all a bad start :). Maybe we should chat about plans to go in and mine out some more.

    Looks like a fantastic conference.

    Greg

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